Step 1, 2 work on your old phone and PC (or Mac/Linux), step 3 works on your Android phone plugged on your PC, step 4 works with SMS Migrator on your Android Phone.

Notes
- If your phone is not supported by above software, drop me a few lines with the detail of your phone (such as phone model, OS, etc.)
- If you use other software to export your SMS, please contact me.